I K J K J. Chapter 1. Problem Solutions. Semiconductor Physics and Devices: Basic Principles, 3 rd edition Chapter 1

Semcouc hyscs evces: sc rcles, r eo her. () cc: 8 cer oms /8 om 6 ce oms ½ oms ol o 4 oms er u cell () cc: 8 cer oms /8 om eclose om om ol o oms er u cell mo: 8 cer oms /8 om 6 ce oms ½ oms 4 eclose oms 4 oms ol o 8 oms er u cell. () 4 G oms er u cell 4 esy 565 0 8. esy o G.0 4 As oms er u cell, so h esy o As.0 () 8 Ge oms er u cell 8 esy 565 0 8. esy o Ge 4.440. () Smle cuc lce; r U cell vol r 8 r om er cell, so om vol. () G 4πr 4πr o 00% 8r () ce-ceere cuc lce 4r c h o 5.4% r U cell vol r 6 r her 4 oms er cell, so om vol. G πr r 4 4 π o 00% 6 r oy-ceere cuc lce 4 4r r U cell vol. r 4 4 4 o 74% oms er cell, so om vol. G πr 4 r 4 π o 00% o 68% 4r () mo lce 8 oy ol 8r r U cell vol. 8r 8 oms er cell, so om vol. 8 4 r 8 4 π o 00% 8r πr o 4%.4 rom rolem., erce volume o cc oms s 74%; heree er coee s rou, olume 074.

Semcouc hyscs evces: sc rcles, r eo her.5 () 54. A rom., 8 r 54. so h r 8. A 8 8 eer o oe slco om o ceer o eres eh r.6 A () umer esy 8 esy 50 54 0 8. ss esy ( A. W. ) 50 ( 809. ) ρ 60. 0 A ρ. rms /.6 () r ( 0. ).04 A A r + r r.04.04 A so h r 0747. A () A-ye; om er u cell esy.04 0 8.7 () esy(a) 8. 0 -ye: om er u cell, so esy() 8. 0 8. + 0..8 A : esy.80.8 0 8 l: esy (sme s ) () : A.W..99 l: A. W. 5.45 So, mss er u cell +.99 5. 45 60. 0.80 4.85 0 mss esy s ρ.8 () so h 4.850.8 0 8 ρ. m /. + 8. 8 A 4.6 A esy o A esy o () Sme s () Sme merl 4.6 0 8 4.6 0 8.9 () Surce esy 8 4.60. 0 4 Sme A oms oms () Sme s () Sme merl.0 () ol esy o Surce esy () Sme s (). Skech. () (),,,, 4 4 o ( ) ( ) 0. 0 0. 0 4

Semcouc hyscs evces: sc rcles, r eo her. () sce ewee eres (00) les s: 56. A ()sce ewee eres (0) les s: 56. 98. A sce ewee eres () les s: 56. 5. A.4 () Smle cuc: 4.50 A () (00) le, surce esy, om 4.940 4.50 0 8 4 () (0) le, surce esy, om 49. 0 4.50 0 8 4 () () le, surce esy, oms 6 () c h.850 4.500 8 4 () oy-ceere cuc () (00) le, surce esy, Sme s (),(); surce esy 4.940 4 () (0) le, surce esy, oms 699. 0 4.50 0 8 4 () () le, surce esy, Sme s (),(), surce esy.850 4 ce ceere cuc () (00) le, surce esy oms 9.880 4.50 0 8 4 () (0) le, surce esy, oms 699. 0 4.50 0 8 4 () () le, surce esy, + 6 4. 0 5 4 4.500 8.5 () (00) le o slco smlr o cc, oms surce esy 54 0 8. 6. 780 4 () (0) le, surce esy, 4 oms 9.590 5 4 0 8. 4 () le, surce esy, 4 oms 7.80 4 5 4 0 8..6 4r he 4r 4.5 664. A () 4 oms olume esy 664 0 8. 55. 0 () sce ewee (0) les, 664. 5

Semcouc hyscs evces: sc rcles, r eo her 4.50 A Surce esy oms 664. 0. 490 4 8.7 esy o slco oms 50 4 vlece elecros er om, so esy o vlece elecros 0.8 esy o GAs oms 8 oms 4.440 8 565. 0 A vere o 4 vlece elecros er om, esy o vlece elecros 77. 0.0 6 50 ( 098. ) () rco y weh 50 ( 806. ) 0. 0 6 () rco y weh 8 0 ( 0. 8) 6 50 0. 98 + 50 8. 06 7.70 6. olume esy 0 So 6 7.940 794 A We hve 54. A So 794 46 54. 5.9 0 () ercee 50 40 5 % 0 () ercee 50 0 6 % 6 5 00% 00% 6

Semcouc hyscs evces: sc rcles, r eo her her. omuer lo. omuer lo. omuer lo.4 π rolem.; hse ω cos λ π ω 0 v ω λ λ π + π rolem.; hse + ω cos λ π + ω 0 v ω λ λ π.5 hc hc hν λ λ Gol: 4. 90 e ( 4. 90). 60 9 J So λ 4 0 665. 0 0 4.90 6. 0 λ 054. µ m 9.540 5 esum: 90. e 90. 6. 0 J So λ 4 0 665. 0 0 90. 6. 0 λ 0654. µ m 9 654. 0 5.6 () lecro: ().. e 6. 0 9 J 9 m 9.0 6. 0 54. 0 5 k m/ s 4 h 665. 0 λ 5 54. 0 λ. A ().. 00 e. 60 7 J m 54. 0 4 k m/ s λ h λ. A () roo:.. e 6. 0 9 J 7 9 m 67. 0 6. 0.0 k m/ s 4 h 665. 0 λ.0 λ 087. A use Aom: A. W. 8.9 e 6. 0 9 J m 7 9 89. 66. 0 6. 0. 0 k m/ s 4 h 665. 0 λ. 0 λ 00. A () A 000 k rvel 0 m/s: mv ( 000)( 0 ) 40 4 k m/ s 4 h 665. 0 λ 4 40 λ 66. 0 8 A 9

Semcouc hyscs evces: sc rcles, r eo her.7 k v 0059. 0. 077 e v m v v (. ). 9 9. 0 0 077 6 0 7.0 6 k m/ s v 4 h 665. 0 λ 6 7.0 λ 9. A.8 hc hν λ h e e e m λ e Se λ λ e 0 e hc h h λ m λ m λ e 0 whch yels h λ 00 mc So e J hc hc mc mc λ 00h 00 8 9.0 0 00 5 64. 0 J 0. ke.9 () mv 9.0 0 4 h m G λ e 80. J 4. 0 e Also mv 9. 0 0 4 8. 0 6 k m/ s 4 h 665. 0 λ 6 8. 0 λ 64 A () 4 h 665. 0 0 λ 5 0 5. 0 6 k m/ s Also 6 5. 0 4 v 58. 0 m/ s m 9.0 v 58. 0 6 / s 4 mv 9.0 58. 0 540. J 9.640 e 4 8.0 hc 665. 0 0 () hν 0 λ 0 99. 0 5 J 5 9 e 99. 0 6. 0 so.40.4 k 5 () m 9.0 99. 0 60. 0 k m/ s 4 h 665. 0 λ λ 0. A 60. 0 0

Semcouc hyscs evces: sc rcles, r eo her. 4 h 054. 0 () 6 0 054. 0 8 k m/ s () hc hc c λ h So c( ) 0 8 054. 0 8 6. 0 0 J 098. e. 4 () h 054. 0 0 0 878. 0 6 k m/ s () 6 m 878. 0 9 50 7.70 J 4.80 4 e. () Sme s. (), 878. 0 6 k m/ s () 6 m 878. 0 6 50 7.70 6 J 4.80 7 e.4 4 h 054. 0 054. 0 0 054. 0 mv v m 500 v 70 6 m/ s.5 4 h 054. 0 () 0 0 054. 0 4 k m/ s 4 () 054. 0 9 () 6. 0 66. 0 6 s.6 () Ψ (, ) Ψ, Schroer s wve equo, he re soluos o h Ψ, Ψ, + () Ψ (, ) jh m h Ψ (, ) Ψ, + () Ψ (, ) jh m A he wo equos, we o h Ψ(, ) + Ψ (, ) m + Ψ, + Ψ, jh Ψ(, ) + Ψ (, ) whch s Schroer s wve equo. So Ψ(, ) + Ψ (, ) s lso soluo. () Ψ Ψ were soluo o Schroer s wve equo, he we coul wre h Ψ Ψ+ () Ψ Ψ m jh Ψ Ψ whch c e wre s h + + Ψ Ψ Ψ Ψ Ψ Ψ m Ψ Ψ + Ψ Ψ jh Ψ + Ψ v y Ψ Ψ we h () + Ψ Ψ Ψ Ψ + m Ψ Ψ ΨΨ () + Ψ Ψ jh + Ψ Ψ

Semcouc hyscs evces: sc rcles, r eo her Sce Ψ s soluo, he h Ψ + () h Ψ j m Ψ Ψ Surc hese ls wo equos, we re le wh h Ψ Ψ Ψ + m Ψ ΨΨ Ψ jh Ψ Sce Ψ s lso soluo, we my wre h Ψ Ψ + () jh m Ψ Ψ Surc hese ls wo equos, we o h Ψ Ψ () 0 m ΨΨ hs equo s o ecessrly vl, whch mes h ΨΨ s, eerl, o soluo o Schroer s wve equo..7 Ψ, As π e jω z + + Ψ, A s π z + A ( ) s π 4π whch yels A A +,, + j, j.8 Ψ, As π e jω z + + 0 Ψ, A s π + A ( ) s π 4π 0 whch yels A z 0.9 oe h * z 0 Ψ Ψ uco hs ee mlze () o 4 z e 0 z 4 o o o e o 0 o o o e o o 0 o o e e 4 whch yels 09. () o z e o 4 z o 4 o o o e o o o e o J e e whch yels 09. o z e z 0 o o o o e o 0 o o e whch yels 0865. J 4 o o 4 o e o o 0 A +,, + j, j

Semcouc hyscs evces: sc rcles, r eo her.0 () k ω cos k ω 0 v + ω k v 5. 0 m s 50 4 0 / 9. v s 06 / () π π π k λ λ k 5. 0 9 λ 49. A Also 4 h 665. 0 0 λ 49. 0 58. 0 5 k m/ s hc 665. 0 0 hν 0 λ 49. 0 4 8 4.740 7 J.960 e. ψ() A jk+ ω where k m h e 9 9.0 005. 60. 054. 0 k 67. 0 8 m 005. 6. 0 4 ω h 054. 0 4 9 ω.80 r / s. h 4 π 054. 0 π m 9.0 000 so 608. 0 J 0 76. 0 e 76. 0 e 50. 0 e 8. 0 e. () h π m 4 054. 0 π 0 9.0 0 4.80 0 ( J ) So 0 4.80 J 06. e 9 67. 0 J 04. e () hc hc hν λ λ 4 8 665. 0 0 λ 9 0 67. 0 4.80 λ 59. 0 6 m λ 59. µ m.4 () he e oel well h π m m h π so 5 0 0 0 8. 0 4 054. 0 π 56

Semcouc hyscs evces: sc rcles, r eo her 5. 0 () h π m ( + ) h π ( + ) m 4 8 054. 0 π () 5. 0 5 0 0 48. 0 0 J ery he (+) se s 48. 0 0 Joules lrer h 0 mj. Quum eecs woul o e oservle..5 euro : 4 h π 054. 0 π 7 4 m 66. 0 0 6.060 e elecro he sme oel well: 4 054. 0 π 4 9.0 0 9 76. 0 e.6 Schroer s wve equo ψ () m + ( ()) ψ h We kow h ψ() 0 () 0 + so hs reo ψ () m + ψ () 0 h Soluo s o he m ψ() Acos + s 0 m where h oury coos: ψ() 0 +, So, rs moe: ψ () Acos where π so π h m Seco moe: ψ () s where π so hr moe: ψ () Acos where π so ourh moe: ψ 4 () s where 4π so 4 4 π h m 9 π h m 6π h m.7 he - wve equo cres coes, (,y,z) 0 + + ψ(, y, z) ψ(, y, z) ψ(, y, z) y z m + ψ (, y, z) 0 h Use sero o vrles, so le ψ(, y, z) X()()() Y y Z z Susu o he wve equo, we e X Y Z m YZ + XZ + XY + XYZ 0 y z h m v y XYZ le k, we h o X Y Z () + + + k 0 X Y y Z z We my se 4

Semcouc hyscs evces: sc rcles, r eo her X X k so + k X 0 X Soluo s o he m X() Ask + cosk oury coos: X() 0 0 0 π X( ) 0 k where,,,... Smlrly, le Y Z k y k z Y y Z z Aly he oury coos, we π y k y, y,,,... π z k z, z,,,... rom quo () ove, we hve k k k + k 0 y z m k + k + k k y z h so h h π m + + y z y z.8 he -mesol e oel well: + ψ, y ψ, y m + ψ(, y) 0 y h e ψ(, y) X()() Y y susu, X Y m Y + X + XY 0 y h ve y XY So X Y m + + 0 X Y y h e X k X X + k 0 X Soluo s o he m: X Ask + cosk u X( 0) 0 0 So X Ask Also, X( ) 0 k π Where,,,... π So h k We c lso ee Y k y Y y Soluo s o he m Y sk y + cos k y y y u Y( y 0) 0 0 Y( y ) 0 k π y y so h π y k y m k k + 0 y h whch yels h y y + m π J Smlres: eery s quze erece: ow uco o eers.9 () ervo o eery levels ecly he sme s he e. h π () m, h π m 5

Semcouc hyscs evces: sc rcles, r eo her () 4 A 4 054. 0 π 7 0 67. 0 40 85. 0 e () 05. 4 054. 0 π 7 67. 0 0. 50.460 7 e.0 () reo, > 0 ψ () m + ψ () 0 h Geerl m o he soluo s ψ () A e j + e j where m h erm wh rereses ce wve, erm wh A rereses he relece wve. eo, < 0 ψ () m + ψ () 0 h he eerl soluo s o he m ψ () A ej + ej where m h erm volv rereses he rsme wve, he erm volv A rereses he relece wve; u rcle s rsme o reo, wll o e relece so h A 0. ψ () ej ψ () A ej + ej () oury coos: () ψ 0 ψ 0 () () ψ () ψ 0 0 Aly he oury coos o he soluos, we A + A om hese wo equos, we A + he releco coece s * AA * + he rsmsso coece s 4 +. reo, > 0, we hve ψ () A e where m h.4 e. e S + 9 9.0.4. 60. 054. 0 4 9.80 m roly comre o 0, ve y ψ () e ψ () 0 () A 9 0 e.80 0 8. 0 08. % () 48 A 9 0 e.80 480 9. 0 0 %. 6 e,. e We hve h U W / 6

Semcouc hyscs evces: sc rcles, r eo her G 6 e where m h / 9 9.0 ( 6.) 60 S 4 0540.. 9.980 m 0 0 m 9 U W 9 0 e 6. 6 6 9.980 0 050. 0 9 m 7.970 9. Assume h quo [.6] s vl: G 6 e () m 0067. S m h m o 9 U 4 054. 0 W 0067 (. ) 9.0 ( 08. 0. ) 60. / 9 07. 0 m 6 0. 0. e 07. 0 50 08. 08. 08. () m ( 08. ) S 9 0 m o 9 08. 9.0 0. 8 0. 6. 0 054. 0 4.40 m 9 4 U W / 4.40 50 9 0 e 7. 0 5.4 6 e 6 e 4 0 0, 0, 0 m m 67. 0 7 k m h 7 6 9 67. 0 ( 0 ) 0 6. 0 S 4 054. 0 4 580. 0 m So 6 e 580. 0 0 0 0 06. 0 5 U W / 4 4.5 eo, 0 ( < 0 ); eo, ( 0 < < ) ; eo, 0 ( > ). () eo ; ψ () A ej + ej (ce) (relece) eo ; ψ () A e + e eo ; ψ () A e j + ej () reo, he erm rereses relece wve. owever, oce rcle s rsme o reo, here wll o e relece wve whch mes h 0. oury coos: 0: ψ ψ A + A + ψ ψ j A j A : ψ ψ A e + e A ej A lso 7

Semcouc hyscs evces: sc rcles, r eo her ψ ψ Ae e j A ej rsmsso coece s ee s * AA * AA so rom he oury coos, we w o solve A erms o A. Solv A erms o A, we A ja + e e 4 l s j e + e e j We he h l + 4 e+ e r m * * AA AA e 4 e We hve h sce >>, he wll e lre so h e>> e we c wre AA * l +4 * AA e 4 whch ecomes e r * * AA AA + e 4 Susu he eressos, we m + h m m h h h m J h * m AA e * h m 6 J h * AA G 6 e AA AA * G 6 m AA lly e *.6 eo : 0 ψ m + ψ 0 h ψ A ej + ej (ce wve) (relece wve) m where h eo : ψ m + ψ 0 h ψ A e j + ej (rsme (relece wve) wve) m where h eo : ψ m + ψ 0 h ψ A e j (rsme wve) m where h here s o relece wve reo. 8

Semcouc hyscs evces: sc rcles, r eo her he rsmsso coece s ee s * * v AA AA * * v AA AA rom oury coos, solve A erms o A. he oury coos re: 0: ψ ψ A + A + ψ ψ A A : ψ ψ A e j+ e j A ej ψ ψ Aej e j Aej u π ej e j, elm, A, rom he ove equos, we hve 4 4 + +.7 () eo : Sce >, we c wre ψ m ψ 0 h eo : 0, so ψ m + ψ 0 h eo : ψ 0 he eerl soluos c e wre, kee m h ψ mus rem e < 0, s ψ e+ ψ A s + cos ψ 0 where m h m h () oury coos: 0: ψ ψ ψ ψ : ψ ψ A A s + cos 0 A A A sce, he G A rom A, we c wre G G whch ves G ur, hs equo c e wre s m h m h hs ls equo s vl oly secc vlues o he ol eery. he eery levels re quze..8 4 me o 4π h o me o 4π h o ( J) ( e ) 9.0 6. 0 9 4 4 8. 85 0 054. 0 π 58. ( e ) 9

Semcouc hyscs evces: sc rcles, r eo her 58. e. 95e. 5e 4 0849. e 4.9 We hve / ψ 00 π r G e o o * r r r o G 4π ψ ψ 4π e 00 00 o π 4 r r e o o o he mmum roly () r 0 r 4 + r G r G r e r e S o o o o UW whch ves 0 r + r o o r o s he rus h ves he rees roly..40 ψ 00 s eee o θ φ, so he wve equo shercl coes reuces o mo ψ r + ( 0 () r ) ψ r r r h where () r ψ 00 e r h 4π m r π o o o / r G e o o / r o G o G e o ψ 00 r π r ψ 00 r π o so h r ψ 00 r r 5 / r e G r 5 / G G G G G o o o o G / h G o e o o π o o r r r r e e π o o o o Susu o he wve equo, we hve 5 / r r r r e e r π m r + + h mr where 4 me o 4π h o o h m he ove equo ecomes / π r G r e r r + m o h o π S U W o o o o h m / h + 0 mr o o o o S r G e o o UW o + + + 0 r r o o o o whch ves 0 0, shows h ψ 00 s ee soluo o he wve equo..4 All elemes rom Grou colum o he eroc le. All hve oe vlece elecro he ouer shell. 0 0

Semcouc hyscs evces: sc rcles, r eo her her. o were o crese, he eery woul ecrese he merl woul e o ehve less lke semcouc me lke mel. o were o ecrese, he eery woul crese he merl woul e o ehve me lke sul.. Schroer s wve equo h Ψ(, ) Ψ(, ) + () Ψ(, ) jh m e he soluo e o he m Ψ, u e j k h eo, () 0, so susu he roose soluo o he wve equo, we o h { jku e j k m h + () () U u e j k h W j () j h ue j k h h jk u j k h () e j k h () U u e j k h W + () u e j k h whch ecomes h { e m + jk u + hs equo c he e wre s () + + () ku () jk u u m + u () 0 h Se u () u () reo, hs equo ecomes () () u jk u + k α u () 0 where α m h Q... reo, (). Assume he sme m o he soluo Ψ, u e j k h Susu o Schroer s wve equo, we o h { jk u e j k m h () e j k h () U u e j k h W + () u e j k h () u e j k h + jk u + hs equo c e wre s () + + () ku () jk u u m m () + () 0 u u h h Se u () u () reo, hs equo ecomes u() jk u () + where α m h m k α + u 0 h () Q...

Semcouc hyscs evces: sc rcles, r eo her. We hve u () jk u + () k α u () 0 he roose soluo s u () A e j ( α k ) + e j ( α + k ) he rs ervve s u () j ( α k ) Ae j ( α k ) j( α + k) e j( α + k) he seco ervve ecomes u() j( α k) Ae j( α k) + j( α + k) e j( α + k) Susu hese equos o he erel equo, we α k Ae j α k ( α + k) e j( α + k) + jk{ j( α k) A e j( α k) j( α + k) e j( α + k) } k α { Ae j( α k) + e j( α + k) } 0 l k q α Ae j( α k) + m cα + αk + kh + kα + k k αqe j( α + k) om erms, we hve α αk + k k α k We h 0 0 Q... he erel equo u () he roose soluo, he roceure s ecly he sme s ove..4 We hve he soluos u () A e j ( α k ) + e j ( α + k ) 0 < < u ( ) e j ( β k ) + e j ( β + k ) < < 0 he oury coos: u () 0 u () 0 whch yels 0 A+ 0 Also u u 0 0 whch yels ( α k) A ( α + k) ( β k) + ( β + k) 0 he hr oury coo s u () u ( ) whch ves Ae j α k + e j α + k ( β ) e ( β ) e j k + j + k hs ecomes Ae j α k + e j α + k ( β ) e ( β ) 0 e j k j + k he ls oury coo s u u whch ves j( α k) Ae j( α k) j( α + k) e j( α + k) j( β k) e j( β k) j( β + k) e j( β + k) hs ecomes α k Ae j α k ( α + k) e j( α + k) ( β ) e ( β ) ( β ) ( β ) k j k + + k e j + k 0.5 omuer lo.6 omuer lo.7 s α + cosα cos k α e k y, α s + cos cos y oser o hs uco y 4

Semcouc hyscs evces: sc rcles, r eo her { s + cos } s y We o ( )() + () s cos y y S S s y UW y s y + y s cos s s y k π, 0,,,... s y 0 So h, eerl, he ( α) α 0 y ( k) k A / m α m m α h k h h k hs mles h α 0 k π k k.8 α ( α) 9 s + cosα cos k α () k π cos k s o: α π : o: α 66. π ( o y rl err) m α h α h m So ( α) 4 054. 0 0 50 9.0 α.490 0 J α 054. e So α π 504. e α 66. π 4.45 e UW y.64 e () k π cos k + s o: α π o: α.54 π 6065. e 9.704 e 4 so 69. e k π cos k s o: α π o: α 44. π 57. e 5 7.799 e 6 so 4.6 e () k 4π cos k + s o: α 4 π o: α 4.7 π 4.066 e 7 8. 74 e 8 so 4.66 e.9 () 0 < k < π k 0 cos k + y rl err: s o: α 08. π o: α π rom rolem.8, α 054. e 06. e 504. e so 0488. e () π < k < π Us resuls o rolem.8 s o: α 66. π o: α π 5

Semcouc hyscs evces: sc rcles, r eo her 4.45 e 6065. e 4 so 87. e π < k < π s o: α.54 π o: α π 9.704 e 5 57. e 6 so 8. e () π < k < 4π s o: α 44. π o: α 4 π 7 7.799 e 8 4.066 e so 67. e.0 6 s α + cosα cos k α e eery s () k π cos k s o: α π o: α 56. π (y rl err) rom rolem.8, ( α) ( 054. ) e 504. e 660. e so.6 e () k π cos k + s o: α π o: α.4 π 6065. e 8809. e 4 so.79 e k π cos k s o: α π o: α. π 57. e 5 6. 679 e 6 so 4. e () k 4π cos k + s o: α 4 π o: α 4.6 π 4.066 e 7 7.96 e 8 so. e. Allowe eery s Use resuls rom rolem.0. () 0 < k < π s o: α 0759. π (y rl err) o: α π We hve α 054. e 08665. e 504. e so 068. e () π < k < π s o: α 56. π o: α π 660. e 6065. e 4 so.6 e π < k < π s o: α.4 π o: α π 6

Semcouc hyscs evces: sc rcles, r eo her 8809. e 5 57. e 6 so 4.7 e () π < k < 4π s o: α. π o: α 4 π 6. 679 e 7 4.066 e 8 so 7.9 e. 00 ; 4.7 0 ( 00) 70. 66 + 00 64. e 00 47. e 00 5. e 400 097. e 500 066. e 600 0. e. he eecve mss s ve y * m h k We hve h curve A> curve k k so h * * m curve A < m curve 4.4 he eecve mss hole s ve y * m h k We hve h curve A> curve k k so h * * m curve A < m curve.5 os A, : < 0 k velocy reco; os, : > 0 velocy + reco; os A, ; k < 0 os, ; > 0 k.6 k h m A k 0 A So eve eecve osve eecve. 0 0 9 k A + k 0 9 m A: 9 4 0 054. 0 9 ( 007. ) 6. 0 m whch yels so m 4.960 k curve A; m 0544. m o : 9 4 0 054. 0 9 ( 07. ) 6. 0 m whch yels m 4.960 k so urve : m 00544. m o.7 k h m m mss; mss; 7

Semcouc hyscs evces: sc rcles, r eo her * 9 k 0. A 0 m urve A: 9 4 0 054. 0 9 ( 008. ) 6. 0 m whch yels m m 4.40 k 0476. m o urve : 9 4 0 054. 0 9 ( 04. ) 6. 0 m whch yels m m 868. 0 k 0095. m o.8 () hν 9 ( 4. ) 6. 0 ν 4 h 665. 0 ν 4. 0 4 z () 8 c 0 7 λ 875. 0 m 4 ν 4. 0 λ 0875. µ m.9 urve A: ecve mss s cos urve : ecve mss s osve rou k 0, s eve rou k ± π..0 k k ( α) s α k k k + αs αk k So α cos αk k k cos α α k k k We hve m * h k α h m * h α. he -mesol e oel well, () 0 whe 0 < <, 0 < y <, 0 < z <. hs reo, he wve equo s + + ψ, y, z ψ, y, z ψ, y, z y z m + ψ (, y, z) 0 h Use sero o vrles echque, so le ψ(, y, z) X()()() Y y Z z Susu o he wve equo, we hve X Y Z YZ + XZ + XY y z m + 0 XYZ h v y XYZ, we o X Y Z m + + + 0 X Y y Z z h e X X k + k X 0 X he soluo s o he m X() As k + cos k Sce ψ(, y, z) 0 0, he X() 0 0 so h 0. Also, ψ(, y, z) 0, he X() 0 so we mus hve k π, where,,,.. Smlrly, we hve Y Z k y k z Y y Z z rom he oury coos, we 8

Semcouc hyscs evces: sc rcles, r eo her k π k π y y z z where y,,,... z,,,... rom he wve equo, we hve m k k k + 0 y z h he eery c he e wre s h m + + π y z. he ol umer o quum ses he - mesol oel well s ve ( k-sce) y ( kk k ) k π π where m k h We c he wre k m h k he erel, we o k m m h h Susu hese eressos o he esy o ses uco, we o ( m m ) π π h h o h h h m hs esy o ses uco c e smle wre s ( ) 4π ( m ) / h v y wll yel he esy o ses, so h. / 4π m h / * 4π m ( ) h.4 / * + k 4π m h z * / + m k / h 4π * / 4πm ( k) h / / 4π 0. 067 9. 0 4 665. 0 0059 6 0 9 /... 80 m. 80 7 / * 4π m ( ) h.5 () / * 4π m h z k * / m / h k 4π * / 4πm / ( k) h / 4π 048. 9.0 4 665. 0 (. ). 0059 6 0 9 / 6. 90 m 6. 90 4 8 / * 4π m ( ) h / 4π 08. 9.0 9 / 6. 0 4 665. 0 4.770 46 m J 9

Semcouc hyscs evces: sc rcles, r eo her ( ) 7.60 e + 005. e + 00. e + 05. e + 00. e / 7 0. e.4 0.96 0 4. 0 * 4π m () ( ) h / 4π ( 056. ) 9.0 9 / 6. 0 4 665. 0 78. 0 46 m J ( ).850 e ( ) 005. e 00. e 05. e 00. e.6.7 omuer lo / * m * / m.8! 0!!! 8! 0 8! ( 0)( 9)( 8! ) ( 0)( 9) ( 8! )(! ) ()() 067 0. e 090. 0 0. 0 7. 0 m / * * m 45 ().0 + e () + ( ) 069. k e k + e + e k () k, ( ) ( ) 069. () 5 k, ( ) 669. 0 0 k, ( ) 4.540 5 069.. ( ) + e k ( ) + e k + e () + e 5 () + e 0 () k, 0. 69 () 5 k, ( ) 6. 690 5 0 k, ( ) 4.54 0.9 () + e + k k. () 00 k 0. 059 e + e k e k 0

Semcouc hyscs evces: sc rcles, r eo her + ( ) k + k + ( ) k + k 64. 0 5 90. 0 5.60 5 4. 0 5 087. 0 5 () 400 k 0. 045 + ( ) k + k + ( ) k + k ( ) 7.70 4 4.50 4.640 4 60. 0 4 097. 0 4. h 4 π 054. 0 π 0 m 9.0 00 0 J 608. 0 076. e 4 60. e, 4 5 9.40e. 5 As s romo > 0, ssume he roly o 5 se e occue s he sme s he roly o 4 se e emy. 4 5 + e + e k k 4 5 + e + e k k + 4 5 4 5 60. + 9.40 e 7.7.4 () -mesol e oel well, h π y z + + m y z 4 054. 0 π + + 9 9.0 0 076. + + e y z 5 elecros, eery se creso o cos oh elecro y z emy se, so ( 076) + +. 84. e () elecros, eery se creso o cos oh y z elecro emy se, so 076 + +. 87. e.5 he roly o se + e occue s + e + e k k he roly o se e emy s + e k e k + e + e k k + + e k ece, we hve h Q...

Semcouc hyscs evces: sc rcles, r eo her.6 () A eery, we w e + e k k + e k hs eresso c e wre s + e k 00. e k 00. 00. e k + kl ( 00) + 4.6k () A + 4.6k, 4.6k + e + e k k whch yels 0. 00990 0. 0.7 () 65. e, 00, A 650. e () + e 6. 40 % 650. 65. 0059. 950 k 0. 059 950 00 k 0080. e ( ) 4.5% + e 650. 65. 0080. 64. 0 5 0045. 00. 099. 00. + e k 00. + e 00. k 099. whch c e wre s + 00. e 99 k 000. 00. 00. l( 99) k 0. 0659 k l( 99) So 756.8 () ( ) 0. 0004 7.5 7.0 + e 0059. 004%. () A 000 k 0. 086 e ( ) 0496. 7.5 7.0 + e 0. 086 4.96% ( ) 0997. 685. 7.0 + e 0059. 99.7% () A.9,, ( ) + e k ll emerures. e k

Semcouc hyscs evces: sc rcles, r eo her 00. e 9.0 0059.,. 0. 08. e ( ) 08. + e 0059. 08. e 0059. e 08. ( ). 0059. 780 4 () 04. 07. e A, ( ) 07. e e k 0059. so A, e so.40 () A, A 845. 0 k 96. 0 7 e, he e 04. 0059. 00. k 9.0 6 e 0059. 4. 0.. e, So. e e k 0059. ( ) 66. 0 9 () 04. 0. e, A, ( ) 0. e e k 0059. 6 A, e 7.880 8 k 96. 0 7.4 ( ) + e k so ( ) + e k e 04. 0059. e k k e ( ) k k + e k () 0, < e( ) 0 0 > e( + ) + 0 A.4 () A m, G + e + e S:. e, ( ) + e Ge: 066. e, k. 0 059 (. ) 4.070 0 k

Semcouc hyscs evces: sc rcles, r eo her ( ) + e GAs: 4. e, ( ) + e 066. 0 059 (. ).90 6 4. 0 059 (. ) 4. 0 () Us resuls o rolem.5, he swers o r () re ecly he sme s hose ve r ()..4 6 ( ) 0 055. + e k 055. + 6 + e 0 6 k 0 055. + 6 055. 6 e 0 l 0 k k 055. k 46 l0 6.44 A, 005. So 005. + e k l ( 9) k y symmery,, 005., So k l ( 9 ) l ( 9) k () A 00, k 0059. e 0059. l 9 055. e () A 500, k 0. 047 e 0. 047 l 9 054. e 4

Semcouc hyscs evces: sc rcles, r eo her 4 her 4 4. e k ( ) k e 00 400 600 0. 077 0. 045 0058. () Slco () Germum ( ) 00 400 600 7.68 0 4.8 0 9.74 0 4.6 0 0 860. 0 4 8. 0 6 4. e k 9 9 0.8 0 04 0. 4 e.90 k y rl err 4. omuer lo GAs 8. 8. 0 9 57. 0. e 00 8 00 4.4 ( ) G e k So e k k A 00 k 0. 059 e A 00 k 0. 077 e G. k 7 58. 0 00 e 8. 0 00 0059. 0. 077 06. 0. 75e ( 9.9) whch yels 5. e 00, 7 5. 58. 0 ( 00) e 0059. 5. 09 4.5 () k e e k k e e e k, o he mmum vlue / e k 0 k / e k hs yels / k / k he mmum vlue occurs k + () e k Q e e k k e 7

Semcouc hyscs evces: sc rcles, r eo her 4 e k o he mmum vlue e k Sme s r (). mum occurs k k 4.6 e k e k where k + 4k + 4k e k k 0 e 4 e 5.. 4.7 omuer lo 4.8 ( A) ( ) ( A) ( ) 00854 A k k A e k e e e A k e. 0. 059 4.9 omuer lo ( A) ( ) 47.5 + 00. e ( 0. 059) 4.0 * m m kl G * 4 m J * * Slco: m 056. m, m 08. m 008. e m * * Germum: m 07. m, m 055. m 00077. e m * * Gllum Arsee: m 0. 48m, m 0. 067 m +008. e m 4. () m kl 4 () G m m * 4 0059 4.. l 06. +0058. e m * J m 4 0059 05.. l 0. 0088. e m 4. m k G l 9 04. 0 k l 0495. k 9.80 G 8

Semcouc hyscs evces: sc rcles, r eo her 4 ( ) k e 00 400 600 4. omuer lo ( e) 0. 077 0. 045 0058. 4.4 e ( ) cos, z z + e k e k z 00085. 007. 0056. m e η so h k η k We c wre so h e e e k k he erl c he e wre s η k e whch ecomes k 4.5 e η η e( ) k 0 k z e z z z + e k e k Q e η so h k η k We c wre + e k z e k e k z ( k) η e ( η) ( k) η 0 We h η e ηe( η) η ( η ) + 0 z0 So ( k) e k 4.6 r m We hve r m * * Germum, 6, m 0. 55m r r ( 6) 9 0 5 055 (. ). so r 5. 4 A he ozo eery c e wre s m m * G S (. 6 ) e 9

Semcouc hyscs evces: sc rcles, r eo her 4 055.. 6 009. e 6 4.7 r m We hve r m * * GAs,., m 0. 067m r r ( ). 05 0067 (. ). r 04 A he ozo eery s 4.8 () m m J * G S (. 6) 0005. e 0 5. 0 4 50 0067. (. 6 ). 4.50 5, > -ye () 4.9 so G kl 5 4.50 G 0059. l 0 5. 0 066. e k e 9 04. 0 e 0. 0059..0 5 Assum. 0. 090. e k e.8 0 8 e 090. 0059..70 4 4.0 () 400 k 0. 045 e / 7 400 4.70 7.40 00 e k 7 05. 7.40 e 0. 045 59. 0 4 Also / 8 400 9 7 0 08. 0 00 4. 05. 7. e 9 7. 08. 0 e 0. 045 ().080 4 G kl 7 4.70 G 0059. l 4 59. 0 076. e 4. 076. 44. e 8 44. 70 e 0059. 9.670 7 40

Semcouc hyscs evces: sc rcles, r eo her 4 4. k e G kl 9 04. 0 0059. l 04. e 5 0. 04. 088. e So e k.8 0 G 9 e 088. 0059. 4.9 0 4 4. () e k 0 05. 5. 0 e 0059.. 0 6 () rom rolem 4., 400.80 400 k 0059. 0. 045 e 00 G kl 6. 0 G 0. 045 l.80 09. e rom () 5. 0 0. 0 6.00 4 rom () 4. ().80. 0 6 50. 0 8 e k 6 05. 8. 0 e 0059.. 0 () rom rolem 4., 400 80 9., k 0. 045 e G kl. 0 G 0. 045 l 9 007. e rom () rom () 8. 0 6. 0.44 8. 0. 0 9 809. 0 6 4.4 slco, 00, η k We c wre 8. 0 0 η 060. / 4

Semcouc hyscs evces: sc rcles, r eo her 4 9 η 04. 0 060. / π π 7.040 8 4.5 Slco, 00, 50 9 We hve η / π 9 9 50.8 0 η / π whch ves η 58. / η. k. 0059. 004. e 4.6 he elecro cocero ( ) ( ) he olzm romo les so * / 4π m e h k * / 4πm e h k k k ee k e ( ) o mmum, se () e k / / 0 e + e / 0 e ( ) whch yels + k k he hole cocero ( ) ( ) rom he e, us he well-olzm romo, we c wre * / 4πm h e k k e k k * / 4π m e h k ee k ( ) e ( ) o he mmum o ( ), se 0. Us he resuls rom ove, we he mmum k 4.7 () Slco: We hve e k We c wre + 0045. e, k 9 0045..8 0 e 0059. 9.80 e ( 4.77).450 7 We lso hve 4

Semcouc hyscs evces: sc rcles, r eo her 4 e.40 k.950 A, we c wre + 95. 0 () k, 0045. e + + 9 040 0045.. e 0059. 5 5 9 04. 0 e( 4.77) 50 + 50 +.4 0 9.0 6 () GAs: Assume 00058. e 7 00058. 4.70 e 0059. 7 4.7 0 e (. 4) 87. 0 6 Assume 0045. e 8 0045. 70 e 0059. 8 70e ( 4.) 4.8 omuer lo 4.9 () Ge: 9.00 6 + 0 + 0 +.950.40 + 50 5.40 5 50 5. 0 4.0 he o level + e k 00. + e 0059. 885. 0 4 A ( ) + e k + k +045. 045. + e + 0059..870 5 4

Semcouc hyscs evces: sc rcles, r eo her 4 4. () 0 5 () 0 5. 0 5 0 5. 0 5 0 6 0 5. 0 6 0.50 4 5. 0 0 () 400 k 0. 045 e 9 9 400.8 0 04. 0 e 00.80 Also + + 50 + 50 +.80 0. 0 4.80 4 0 566. 0 0 (e) 500 k 0. 047 e 9 9 500.8 0 04. 0 e 00 854. 0 + +. 0. 045. 0. 047 Also 50 + 50 + 8540. 49. 0 4 854. 0 4 49. 0 4.890 4. () 0 5 () 6 8. 0 5 0 6. 0 0 6 6 8. 0 6 0 4. 0 4 8. 0 6 () k 0. 045 e 7 8 400 e 4.70 70 8. 0 9 (e) k 00 0 4 9 8. 0 4 0 08. 0 5 0. 047 e 4.70 70 7 8 500 e 00 4. 0. 045 4. 0. 047 44

Semcouc hyscs evces: sc rcles, r eo her 4.80 Also 0 4.80 4 0 7.900 8 4. () > -ye () S:.50 0 Ge: 5. 0 0 5. 0 5. 0 5. 0 7 + G + 5. 0 5. 0 + +.40 GAs: A 6. 0.40 6. 0 77. 0 5. 0 6 8. 0 5. 0 06. 4.4 450 e 9 9 450.8 0 04. 0 00 () () 7. 0 > -ye. 0. 059 450 00 + 5. 0 80 + 5 4 5. 0 80 5 4 + + 70 4 7. 0 4 70 4.0 ol oze mury cocero + 5. 0 + 80 4.5.0 5 0 5. 0 5 0 5. 0 5 > -ye 5 4 7. 0 45

Semcouc hyscs evces: sc rcles, r eo her 4 4.6 k 0059. 00 00 0. 077 e 00 4.7 0 7 0 00 e 7 8 8. 5 \ 5 067. A 5 09. 4.7 omuer lo 4.8 omuer lo 4.9 omuer lo 4. 0. 077 4.40 -ye, so mjy crrer elecros 4.4 () + + 0 + 0 + 0 4. 0 0 4. 0. 0 > -ye 0 0 6 6 () 0 6 0 5. 0 6 0.50 4 > -ye 0 0.8 0 6 0 5. 0 6.80 804. 0 6 5 4.4 () < -ye () 4.50 4 elecros: my crrer 0 5. 0 4 4.50 50 5 holes: mjy crrer so 5 5 50 50 0 6 Acce mury cocero, 50 5 o mury cocero 4.4 G kl Germum: k( e ) 00 400 600 0. 077 0. 0454 0058..6 0 0 86. 0 4 8. 0 6 46

Semcouc hyscs evces: sc rcles, r eo her 4 e + + ( ) 00 400 600 0. 0 5 49. 0 5 87. 0 6 0 5 0855. 0. 0898 0. 000674 4 7 8 504. 0 4.70 7 0 00 y rl err 4.46 omuer lo e 76 4. 0. 059 00 4.44 G kl Germum, 00.40 + + e 0 4 0 6 0 8 05. 0 4 0 6 0 8 4.45 + + 005. so 008. 056. 0755. + + 5. 0 5 5. 0 5 005. whch yels 0075. 0 5 504. 0 4 We hve e k so 4.47 omuer lo 4.48 () m kl 4 G m m * * J ( 0059) ( 0) 4 +00447. e m () mury oms o e e so e m 045. () -ye, so cce mures () 0. 0447 + 0. 45 0. 4947 e e k 5 0 e 97. 0 4.49 k so 5 + 9 05. 50.80 e 0059. so 4.50 e 50 + 6. 950 5 5. 0 6 () k e 04947. 0059. 47

Semcouc hyscs evces: sc rcles, r eo her 4 9 + 04. 0 e 6 045. 8. 0 e 5 k 70 0059. 49. 0 ( 0. 059) l 49. 0 089. e () 089. 0. 059 0. 6 e 9 06. 04. 0 e 0059. so h 90. 0 6 70 5. 0 6 Acce mures o e e 90. 0 6 4.5 5 () k 0 G l 0059. l 0 5. 0 0877. e () kl e 0877. (), 0 5 () 4.5 0 5. 0 5 0.50 5 G kl 0059. l 045. e G so 4.5 6. 0 <, os mus e e 0 5 6. 0 9.680 4 G () kl () () (e) 5 0 0059. l 0 5. 0 G 0056. e G kl 6 0 0059. l 0 5. 0 G 047. e k 0. 045 e,.80 G kl 4 0 0. 045 l.80 G 09. e k 0. 047 e, 8. 540 G kl 4 49. 0 0. 047 l 854. 0 G 0004. e 48

Semcouc hyscs evces: sc rcles, r eo her 4 4.54 () k G l () () 5 0 0059. l 6 8. 0 G 0595. e k G l 6 0 0059. l 6 8. 0 G 058. e k 0. 045 e,. 80 9 4 0 0. 045 l 9 8. 0 G 0565. e (e) k 0. 047 e,.80 G kl 4.55 -ye 4 0 0. 047 l.80 G 056. e G kl 5 50 0059. l 0 5. 0 G 094. e 49

Semcouc hyscs evces: sc rcles, r eo her 4 (e le lk) 50

Semcouc hyscs evces: sc rcles, r eo her 5 her 5 5. () 0 6 6 8. 0 6 0 4. 0 4 () J eµ Ε GAs oe 0 6, µ 7500 / s 9 6 J 6. 0 ( 7500) 0 ( 0) () () J 0 A/ 0 6, 4. 0 4 () GAs oe 0 6, µ 0 / s J eµ Ε 6 0 ( 0) 0 ( 0) 9 6. J 4.96 A/ 5. () 0 ( 0. ) 00 Ω () σ σa A 0 σ ( 00) 0 σ 00(. Ω ) σ eµ 9 00. 6. 0 ( 50) 4.60 () σ eµ 9 00. 6. 0 ( 480) 0. 0 0 4 5. 0 5 oe: he o coceros oe, he ssume moly vlues re vl. 5. ρ () σ eµ A σa 50 6, µ 00 / s 0. 9 6 4 6. 0 ( 00) 50 ( 00) 0 6. 0 4 Ω 5 044. ma 6. 0 4 () hs cse 6. 0 Ω 5 4.4 ma 6. 0 Ε 5 (), Ε 50 / 00. A v µ Ε ( 00)( 50 ) v 55. 04 / s 5 (), Ε 500 / 00. A v 00 500 v 55. 05 / s 5

Semcouc hyscs evces: sc rcles, r eo her 5 5.4 () GAs: ρ 0 05. kω A 0 σa σ eµ 0 7, µ 0 / s σ 6. 0 0 0. 6 Ω So σa ( 500)(. 6) 850 8 4. µ m () Slco 0 7, µ 0 / s σ 6. 0 0 0 4.96 Ω So σa ( 500)( 4.96) 850 8. µ m 5.5 () Ε / v () v 9 7 9 7 v 0 µ Ε µ Ε µ / s µ Ε ( 800)( ) v.4 s 0 / 5.6 () Slco: Ε k /, v. 06 / s 4 0 8. 0 s 6 v. 0 GAs, v 7.5 s 06 / 4 4 0. 0 s 6 v 7.50 () Slco: Ε50 k /, v 9.5 s 06 / 4 0 05. 0 s 6 v 9.50 GAs, v 7 s 06 / 4 0 4. 0 s 6 v 70 5.7 rsc semcouc, σ e µ µ + () 0 4, µ 50 / s, µ 480 / s 9 0 σ 6. 0 5. 0 ( 50 + 480) σ 4.9 0 6 ( Ω ) () 0 8, µ 00 / s, µ 0 / s 9 0 σ 6. 0 5. 0 ( 00 + 0) σ 0. 0 6 Ω 5.8 () GAs σ eµ 5 µ 6. 0 9 rom ure 5., us rl err, we 7. 0, µ 40 / s 54

Semcouc hyscs evces: sc rcles, r eo her 5 8. 0 7. 0.49 0 5 6 () Slco: σ eµ ρ e 9 ρµ 8 6. 0 50 () 579. 0 4 0 5. 0 4 579. 0 89. 0 5 oe: he o coceros oe r (), he ssume moly vlues re vl. 5.9 σ e µ + µ 0 6 9 6. 0 ( 000 + 600) 00 90 9. e k kl 0059. l. e 9 0 9. 0 9. ( 500) 0 e ( 0. 059) 500 00 55. 0 6 500.70 9 σ. 60.70 ( 000 + 600) so σ 500 580. Ω 9 9 0.. 5.0 () () Slco: σ e µ + µ σ 60 50 ( 50 + 480) σ 4.9 0 6 ( Ω ) () Ge: 9 σ. 60.40 ( 900 + 900) σ. 0 ( Ω ) () GAs: 9 6 σ. 60 8. 0 ( 8500 + 400) σ.56 0 9 ( Ω ) () σa 4 () 00 0 6 8 4.9 0 85 0 56. 0 9 Ω 4 () 00 0 8. 0 85 0 06. 0 6 Ω 4 () 00 0 9 8.56 0 85 0 9.90 Ω 5. () ρ 5 eµ Assume µ 50 / s 9 6. 0 ( 50)( 5) 9.60 4 () 00 75 400 5 rom ure 5., 75, 0 5 55

Semcouc hyscs evces: sc rcles, r eo her 5 µ 500 / s 5, 0 5 µ 700 / s Assum 9.60 4 over he emerure re, 00, ρ 9 4 6. 0 ( 500) 9.60 ρ.7 Ω 400, ρ 9 4 6. 0 ( 700) 9.60 ρ 9.64 Ω 5. omuer lo 5. () Ε 0 / v µ Ε v 50 0 v 5. 04 / s so * m v ( 08. ) 9.0 5. 0 7 8 897. 0 J 56. 0 e () Εk /, v 50 000 5. 06 / s 4 ( 08. ) 9.0 5. 0 5.4 () 4 897. 0 J 56. 0 e e k 9 9 e 0 0 0. 0059. 9 9 7.80 8. 470 0 >> 0 4 4 J σ Ε eµ Ε 9 4. 60 ( 000) 0 ( 00) J 60. A/ () A 5% crese s ue o 5% crese elecro cocero. So 4 05. 0 + + We c wre 4. 050 50 50 + so 55. 0 6 whch yels.65 0 9 9 00 e 00 0. e 00 k y rl err, we 456 k 5.5 () σ eµ + eµ eµ σ + eµ o he mmum coucvy, σ eµ ( ) 0 + eµ whch yels G / µ (Aswer o r ()) µ Susu o he coucvy eresso eµ σ σ + eµ µ µ m / µ µ whch smles o σ e µ µ m he rsc coucvy s ee s / 56

Semcouc hyscs evces: sc rcles, r eo her 5 σ σ e µ + µ e µ + µ he mmum coucvy c he e wre s σ µ µ σ m µ + µ () A () A 00, µ ( 00)( 87. ) µ 88 / s 400, µ ( 00)( 0. 65) µ 844 / s 5.6 σ eµ ρ e ρ 50 5 00 ρ 5 50. e 00. e k k k 0059. 0 k 0059. 0. 0849 00 9.05, 7.550 k k ( 9.05 7.550) l ( 0). e 5.7 + + µ µ µ µ + + 000 500 500 0. 00050 + 0. 000667 + 0. 000 µ 6 / s k k 5.8 µ 00 00 00 00 / + / 5.9 + + 0006. µ µ µ 50 500 µ 67 / s 5.0 omuer lo 5. omuer lo 5. J e e () 4 50 0 00. 0 09 6 0 5 50 4 0 9.. 000. ( 09. )( 000. ) 4 50 () 0 9 6. 0 ( 5) whch yels 5. () 0 0 50 4. J e e G 6 5 9 6. 0 5 J A 005. 06. A/ 0 0 0 00. () AJ 005. 06. 8 ma 57

Semcouc hyscs evces: sc rcles, r eo her 5 5.4 so J e e 9 400 6. 0 400 6 5.5 J 5 / s e 0 60 4 0 40 9 6 7 6 e 6 0 e 6. 0 0 0 00 4 0 J 6 A/ cos ll hree os 5.6 J ( 0) e 0 5 9 5 0. 60 ( 0) 0 e 4 50 J ( 0). A/ J ( 0) e 0 50 6. 0 ( 5) 50 e 0 J ( 0) A/ J J ( 0) + J ( 0). + J 5. A/ 4 9 4 6 5.7 J e e 0 5 e.5 sce s µm, so.5.50 4. 5 J e 0 e 4.50.5 5.8 9 5 6. 0 ( 48) 0 e 4.50.5 4. e A/.5 + J J eµ Ε + e 40 6. 9 6 0 960 0 e 8 + Ε 9 6 6. 0 5 0 4 8 0 e 8. e Ε. e 8 8. e 40 8 56. e 8 + Ε 4.5 6 e 8 40 56 Ε 5.9 J J + J, r, () J e, () 0 5 e where µ m so 5 J e e, 0 58

Semcouc hyscs evces: sc rcles, r eo her 5 9 5. 60 ( ) 0 J e, 4 0 00 54. 0 4 A+ e. 0 7 e J + 6. e A/, hs equo s vl ll, so () 00 54. 0 4 A J J J, r, A 65. 0 5 Also J 4.8 6. e r, 54. 0 4 e J eµ Ε 0 7. r, e 0 9 6 6. 0 ( 000) 0 Ε whch yels.50 4.8 6. e A 0, eµ () 0 Ε 50 whch yels so h 9 50. 60 ( 8000)( )( A+ ) Ε e / whch yels 4. 0 5 5.0 5 5 () J e () + e () µ Ε () 65. 0 4. 0 e () 5 5 µ 8000 / s so h A 0, () 0 6. 50. 40 ( 0 059)( 8000) 07. / s r 0 60 5 (). 9 00 6. 0 ( 8000)( ) ( ) A 50 µ m, 9 () 5 5 50 + 6. 0 ( 07) ( 50) 6. 50. 40 e 5. whch yels 4 7 () 50 6 80 5. 00 54. 0 () +. 0 Soluo s o he m A 50 µ m, J eµ ( 50) Ε r () 9 5 A+ e. 60 ( 8000) 68. 0 ( ) so h () e J ( 50) 94.9 A/ r J ( 50) 00 94.9 Susu o he erel equo, we hve J ( 50) 5. A/ 59

Semcouc hyscs evces: sc rcles, r eo her 5 5. e k () +, 04. so h 05. 0 + 04..5 0 04..50 So e G 04..5 0 k () J e.50 0. 4.50 G e e k k Assume 00, k 0. 059 e, 5. 0 0 9 0 J 6. 0 5 5. 0.5 0 0059. e G 04..50 0059. G J 04.5 0 5790 4.. e 0059. () A 0, J.950 A/ () A 5 µ m, J 7. A/ 5. () J eµ Ε + e 9 6 80 6. 0 000 0 Ε + G 6 9 0 6. 0 5. 9 4 where 00 0 We 80 6. Ε 6. Ε 444. 0 80 6. Ε + 444. Solv he elecrc el, we 8. 56 Ε () J 0 A/ 0. 6 Ε + 444. 44. Ε 5. () J eµ Ε + e e o e ( α), J 0 0 µ e α Ε + α e α o o 0 Ε + α Sce So µ k µ e Ε α k e () /α z Ε 0 k α e α / k α z e 0 α k so h e 5.4 rom mle 5.5 9 ( 0059. ) Ε 0 ( 0059. ) 0 6 9 0 0 0 z 4 0 4 0 Ε ( 0059. ) 0 0 0 0 z 60

Semcouc hyscs evces: sc rcles, r eo her 5 0059. 0 l 0 0 ( 0059. ) l ( 0. ) l ( ).7 m 5.5 rom quo [5.40] () () 4 0 k Ε e () 000 ( 0. 059) () () 4 + 86. 0 () 0 Soluo s o he m () A e( α ) () Aα e ( α ) Susu o he erel equo Aαe α 4 + 86. 0 Ae α 0 whch yels α 86. 0 4 A 0, he cul vlue o () 0 s rrry. 5.6 () J J + J 0 r J e e () e ( ) e o We hve k µ ( 6000)( 0. 059) 55. 4 / s e J 9 6 6. 0 ( 55. 4) 5 0 e 4 0. 0 J 4. 0 5 e A/ 0 () 0 J + J r J eµ Ε r 9 6 6. 0 6000 50 e Ε 48Ε e We hve J J r so 48Ε e 4. 0 5 e whch yels 5.7 omuer lo Ε.580 / 5.8 k () µ ( 95)( 0. 059) e so. 96 / s () 8. / s 8. µ µ 09 / s 0059. 5.9 We hve 0 0 m, 4 5 W 0 0 m, 0 0 m () We hve 0 0 m 6 z e.9 m, ma 0 A 0 5. 0 9 5. 60 0 0 6

Semcouc hyscs evces: sc rcles, r eo her 5 () 5.40 () ().90 W 0 09. / z e 05. m 05. 0 W 0 µ 6 500 50 9 5 50 60. 50 56. 0 / e W 6 500 9 4 5 6. 0 50 ( 0. ) 0 50 µ 0. 5 m / s 5 / s 5.4 () osve -ye () z z e e 075. 0 0 9 5 6. 0 58. 0 0 8. 080 m 8. 080 e W µ 5 075. 0 0 9 4 5 6. 0 808. 0 ( 5) 0 0 µ 87. 0 m / s 87 / s 5.4 () W 6. 50 50 085. m () eve -ye z e 0. 50 6. 50 9 5 6. 0 50 0. 850 4.90 m 4.90 () µ e W 5 05. 0 05. 0 9 4 5 6. 0 4.90 (. 5) 50 50 µ 00. m / s 00 / s 5.4 () eve -ye z () 868. 0 4 e µ µ 88 / s e W 9 4 () σ eµ. 60 ( 88) ( 8. 680 ) ρ ρ 088(. Ω ) 6

Semcouc hyscs evces: sc rcles, r eo her 6 her 6 6. -ye semcouc, low-jeco so h δ 50 6 τ 0 50 9 s 6. () τ 0 0 4 0 6 0 4 0 s 7 0 50 0 () δ 0 7 τ 0 50 8 s so 8 0 50 50 50 8 s 6. () ecomo res re equl τ τ 0 6 0 5. 0.50 6 0 So 6 4 0.50 6 τ 00 + τ 889. 0 6 s 4 () Geero e ecomo e So 4.50 G G 5. 0 9 s 6 00 G 5. 0 9 s 4 8 6.4 hc 665. 0 0 () hν 0 λ 6000 5. 0 9 J hs s he eery o hoo. W J / s 7. 0 8 hoos/s + olume ()( 0. ) 0. 8 7. 0 0. 7. 0 9 eh rs/ s () 9 6 δ δ τ 7. 0 00 δ δ 7. 0 4 6.5 We hve + + τ J eµ Ε e he hole rcle curre esy s J + µ Ε ( + e) + µ ( Ε ) We c wre ( Ε) Ε + Ε so 65

Semcouc hyscs evces: sc rcles, r eo her 6 + µ ( Ε + Ε) ( + ) µ Ε Ε + + τ We c he wre µ ( Ε + Ε ) + τ 6.6 rom quo [6.8] + + τ sey-se, 0 + 0 + oe-mesol cse, + 0 0 0 9 + 80 9 s 6.7 rom quo [6.8], + 0 + 00 9 + 0 9 s 6.8 We hve he couy equos () ( δ ) µ Ε ( δ) + Ε + ( δ ) τ () ( δ ) + µ Ε ( δ) + Ε + ( δ) τ y chre eurly δ δ δ δ δ ( δ) ( δ) ( δ) ( δ ) Also, τ τ we c wre () ( δ ) µ Ε ( δ) + Ε + δ () δ + µ Ε δ + Ε + δ ully quo () y µ quo () y µ, he he wo equos. We µ + µ ( δ) + µµ ( ) Ε ( δ) + µ + µ µ + µ ve y µ + µ, he µ + µ ( δ) µ + µ J µµ ( ) + Ε δ µ + µ ee µ + µ µ + µ µµ µ µ + µ + δ ( ) + + δ we hve + + ( δ) ( δ ) µ Ε ( δ) ( ) Q... 66

Semcouc hyscs evces: sc rcles, r eo her 6 6.9 Ge: 00,.40 + + 0 + 0 +.4 0 6. 0 Also.40 6. 0 6. 0 We hve µ 900, µ 900 0, 49. ( + ) + 0 49. 60. + 60. ( 0) 60. + ( 49.) 60. 58. 4 / s Also µµ ( ) µ µ + µ 900 900 6. 0. 60 ( 900). 60 + ( 900) 6. 0 µ 868 / s. 60 6. 0 τ τ τ 4 µ s whch yels τ 54 µ s 6.0 σ eµ + eµ Wh ecess crrers rese +δ +δ -ye semcouc, we c wre δ δ δ σ eµ + δ + eµ + δ σ eµ eµ e µ µ δ so σ e µ µ δ + + + + sey-se, δ τ So h σ e µ + µ τ 6. -ye, so h my crrers re holes. Um eero hrouhou he smle mes we hve δ ( δ) τ omoeeous soluo s o he m ( δ) A G e τ J he rculr soluo s ( δ) τ so h he ol soluo s ( δ) τ A G + e τ J A 0, δ 0 so h 0 τ + A A τ τ J e e e eµ e µ µ δ 9 6 6. 0 ( 000) 50 9 7. 60 ( 000 40) 50 0 e τ J δ τ e he coucvy s σ µ µ µ µ δ so σ + + + + + + + 67

Semcouc hyscs evces: sc rcles, r eo her 6 σ 8+ 04. e where τ 0 7 s τ J 6. -ye GAs: σ eµ + µ ( δ) sey-se, δ τ. 9 7 σ. 60 ( 8500 + 400) 0 0 σ 057(. Ω ) he sey-se ecess crrer recomo re 0 s 6. < 0, sey-se, so 7 δ() 0 τ 50 0 () δ 0 50 5. σ eµ + e µ + µ ( δ) 0, δ δ() 0 e τ 9 6 σ 6. 0 50 50 +. 9 5 60 ( 50 + 480) 5. 0 e τ σ 0. 8 + 0. 49 e τ We hve h Aσ AJ AσΕ so 4 0 () 5 0. 8 + 0. 49 e τ ( 00. ) 54 +.0 e τ ma where τ 0 7 s 6.4 () -ye GAs, + + δ ( δ ) ( δ ) µ Ε ( δ ) τ Um eero re, so h ( δ) ( δ) 0, he δ ( δ) τ he soluo s o he m δ τ e τ δ e τ τ () mum vlue sey-se, So ( δ) ( δ) τ τ 0 4 0 0 0 6 s τ 0 4 eerme whch () δ ( 075. ) 0 4 We hve 4 4 075. 0 0 e τ whch yels τ l 9. µ s 075. () δ 05. 0 4 We τ l 069. µ s 05. () δ 05. 0 4 We τ l 088. µ s 05. 6.5 () 5. 0 5 0 0.50 4 68

Semcouc hyscs evces: sc rcles, r eo her 6.50 τ τ 0 τ.50 7 s 4 δ 0 7 τ.50 4.440 0 s ecomo re creses y he c 0 4.440 4.440 9 0 () rom r (), τ.50 7 s 6.6 Slco, -ye. 0 0 7 s δ τ τ e δ 0 e τ 0 7 0 0 e τ A 0 7 s, e 0 6. 0 δ 0 7 0 ( ) δ 7 > 0 7 s, 7 0 δ 6. 0 e τ where τ 6.7 0 7 s () 0 < < 0 6 s δ τ e τ e τ 4 δ 0 e τ 0 0 0 6 4 where τ 0 6 s A 0 6 s δ µ s 0 4 e δ µ s 08650 4. > 0 6 s 6 δ 0 4 0865. 0 e τ () () A 0, δ 0 () A 0 6 s, δ 0865. 0 4 () A, δ 0 6.8 -ye, my crrers re elecros sey-se, ( δ) 0, he () ( δ ) δ 0 τ ( δ) δ 0 Soluo s o he m δ Ae + e + u δ 0 s so h 0. A 0, δ 0 δ 0 e k τ, where µ e 0 059 00.. / s 50 7 9.4 µ m. Q 69

Semcouc hyscs evces: sc rcles, r eo her 6 () J e ( δ ) e 0 e 9 6. 0. 0 e 4 9.40 J.6 e ma/ 6.9 () -ye slco, 0 4 0 5. 0 6.50 4 0 () cess my crrer cocero δ A 0, 0 so h δ 0 0.50 6 () he oe-mesol cse, ( δ ) δ 0 τ ( δ) δ 0 where τ he eerl soluo s o he m δ Ae + e+, δ rems e, so h 0. he soluo s δ e 6.0 -ye so elecros re he my crrers + + δ δ ( δ ) µ Ε ( δ ) τ ( δ) sey se, 0, Ε 0, so we hve ( δ ) δ δ δ 0 τ where τ he soluo s o he m δ Ae + e + 0 >0, 0 he ecess cocero δ mus rem e, so h 0. A 0, δ () 0 0 5, so he soluo s δ 0 e 5 We hve h µ 050 / s, he k µ 050 0. 059 7. / s e τ 7. 80 7 46. 6 µ m () lecro uso curre esy 0 J e ( δ ) 0 e 0 e 5 0 5 9 5 e 0 6. 0 7. 0 4 46. 60 J 094. A/ Sce δ δ, ecess holes use he sme re s ecess elecros, he J ( 0) + 0. 94 A/ () A, J e 5 ( δ ) e0 ( ) e 9 5 6. 0 7. 0 e 4 46. 60 J 044. A/ J +044. A/ 6. -ye, so we hve ( δ ) ( δ) δ µ Ε τ Assume he soluo s o he m δ Ae s 0 70

Semcouc hyscs evces: sc rcles, r eo her 6 ( δ) As e ( s), ( δ ) As e ( s) Susu o he erel equo Ae( s) Ase( s) µ Ε Ase( s) 0 τ s µ Ε s 0 τ v y µ s Ε s 0 he soluo s s µ µ 4 s Ε ± Ε + hs c e rewre s J µ Ε µ Ε s ± + We my ee µ Ε β s β ± + β er h δ 0 > 0, use he mus s > 0 he lus s < 0. he soluo s δ() > 0 δ() Aes < 0 + where s ± + β β ± 6. omuer lo J 6. () rom quo [6.55], ( δ ) ( δ) δ + µ Ε τ ( δ ) µ ( δ ) + Ε δ 0 We hve h k µ e so we c ee 0 µ Ε Ε k e we c wre ( δ) ( δ) δ + 0 Soluo wll e o he m δ δ() 0 e ( α) where α > 0 ( δ) ( δ) αδ ( ) α ( δ) Susu o he erel equo, we hve δ α ( δ) + α ( δ) 0 α α 0 whch yels α S oe h Ε () + + U W 0,, he α τ where µ e 00 0 059.. / s 7. 50 9.4 m µ Ε /, he 7

Semcouc hyscs evces: sc rcles, r eo her 6 k e 0059. 6. 0 4 Ε α 575. 0 ce o he elecros ue o he elecrc el s he eve -reco. heree, he eecve uso o he elecros s reuce he cocero ros o ser wh he le elecrc el. 6.4 -ye so he my crrers re elecros, he + + δ ( δ ) ( δ ) µ Ε ( δ ) τ Um llumo mes h δ δ 0. τ, we re le wh ( δ) whch ves δ + < 0, δ 0 whch mes h 0. δ G 0 >, 0 so we hve ( δ ) 0 r δ G (o recomo) 6.5 -ye so my crrers re holes, he + δ δ ( δ ) µ Ε ( δ ) τ ( δ) We hve τ, Ε 0, 0 (sey se). we hve ( δ ) ( δ) 0 + < < +, G cos. ( δ) G + G δ + + < <, 0 so we hve ( ) so h δ δ 0 δ + 4 < <, 0 so h ( δ) ( δ) 0,, 5 δ + 5 6 he oury coos re () δ 0 + ; () δ 0 ; () δ couous + ; (4) δ couous ; he lu mus e couous so h ( δ) ( δ) (5) couous + ; (6) couous. Aly hese oury coos, we G δ 5 < < + G δ G δ ( ) < < ( + ) < < 6.6 075. µ 875 / s Ε.5 6 600 µ Ε ( ) 6.5 875 755. 0 6 6600 whch ves 48 9. / s rom he se relo, k µ e 48. 9 875 0. 0608 6 7

Semcouc hyscs evces: sc rcles, r eo her 6 6.7 / Assume h, 4π e 4 s he soluo o he erel equo G o rove: we c wre / G 4π e 4 4 / G 4π e 4 4 / + G 4π e 4 4 Also / 4π e 4 4 G / + / G 4π e 4 Susu he eressos o he erel equo, we 0 0, Q... 6.8 omuer lo 6.9 -ye δ δ τ 0 0 0 We hve 0 6 6 5 0 5. 0.50 6 0 + kl G δ 6 5 0 + 0 G 0059. l 0 5. 0 0498. e 4 k 6.0 () -ye + l G δ 4 5.50 + 0 G 0059. l 0 5. 0 0877. e G kl 5 50 G 0059. l 0 5. 0 094. e () δ δ 50 4 0 5. 0 4 4.5 0 5 50 k + l G δ 4 4 4.50 + 50 G 0059. l 0 5. 0 0697. e k + l G δ 5 4 50 + 50 G 0059. l 0 5. 0 08. e 7

Semcouc hyscs evces: sc rcles, r eo her 6 6. -ye GAs; 50 6 6 8. 0 5 648. 0 6 50 We hve δ δ ( 0. ) 50 5 () + kl G δ 6 5 50 + 50 G 0059. l 6 8. 0 065. e We hve G kl 6 50 G 0059. l 6 8. 0 068. e 0. 65 0. 68 so 0005. e () + kl G δ 5 50 G 0059. l 6 8. 0 056. e 6. Qus-erm level my crrer elecros k We hve 4 δ G 0 µ m 50 + l G δ elec he my crrer elecro cocero kl We 4 () 0 6 m Q 50 µ 8. 0 ( µ m) e 0 0 0 50 058. +06. +079. +040. +048. +046. Qus-erm level holes: we hve + kl G δ We hve 0 6, δ δ We µ m 6. () We c wre kl ( e) 0 50 G +0. 585 +0. 5840 + kl G δ so h ( ) + δ G k l k l + δ kl k ( 00. ) + δ e ( 00. ) 00. 74

Semcouc hyscs evces: sc rcles, r eo her 6 δ 000. low-jeco, so h δ 50 () k G l δ 50 G 0059. l 0 5. 0 0505. e 6.4 omuer lo 6.5 omuer lo 6.6 () + + + τ τ 0 + + τ + + τ τ + τ () We h ee he e eero re s + + where sce hese re he herml equlrum eero recomo res. 0, he τ + τ +. hus eve τ + τ recomo re mles e osve eero re. so h 6.7 We hve h + + + τ + + τ + +δ +δ, he + δ + δ τ + δ+ + τ + δ+ + δ + + ( δ ) + + + + + τ δ τ δ <<, we c elec he ( δ) ; lso δ δ + τ + + τ + () -ye, >>, >> + 0 7 s δ τ () rsc, δ τ + τ 7 7 δ τ + τ 0 + 50 s δ + 67. 0 6 -ye, >>, >> 7 δ τ 50 0 s + 6 75

Semcouc hyscs evces: sc rcles, r eo her 6 6.8 () rom quo [6.56], ( δ ) δ 0 + τ Soluo s o he m δ τ + Ae + e + A, δ τ, so h 0, δ τ + A e We hve ( δ ) s 0 ( δ ) 0 We c wre ( δ) A 0 ( δ) 0 τ + A A s τ + A Solv A we s A τ + s he ecess cocero s he δ τ where s + s e J 7 τ 0 0 0 J 7 s δ 0 0 e 0 0 + s s 4 δ 0 e 4 0 + s () s 0, δ () s 000 / s, 0 4 4 δ 0 067. e J J Q 4 () s, δ 0 e () () s 0, δ 0 0 4 () () J () s 000 / s, δ 0 080 4. () s, δ() 0 0 6.9. 7 4 τ 5 50 5 40 () 7 5 A 0 0 50 0, τ r δ τ 0 5 > 0 ( δ ) δ ( δ) δ 0 0 τ Soluo s o he m δ Ae + e+ A 0, δ δ A+ A W, δ 0 Ae W + e + W Solv hese wo equos, we W A δ e + ew δ ew Susu o he eerl soluo, we δ δ e+ W ew ke + ( W ) e ( W ) δ sh ( W ) δ sh W where δ 0 5 5. 4 µ m () τ, we hve ( δ) 0 so he soluo s o he m δ + 76

Semcouc hyscs evces: sc rcles, r eo her 6 Aly he oury coos, we δ δ W 6.40 τ, we hve ( δ) δ 0 so h A δ A + A W δ s ( ) W δ W A saw ( + ) whch yels A + sw s A 0, he lu o ecess holes s 0 9 δ 0 A so h A 9 0 8 4 0 0 8 0 8 0 0 + sw 0 +W s s he soluo s ow 8 0 δ 0 W + s () s, 8 4 δ 0 00 () s 0 / s δ 0 700 8 4 6.4 W < < 0, ( δ ) G 0 + so h ( δ) G + G δ + + 0 < < W, ( δ) ( δ) 0, so, δ + 4 he oury coos re: () s 0 W, so h ( δ ) W 0 () s + W, so h δ( W) 0 () δ couous 0 ( δ) (4) couous 0 Aly he oury coos, we GW GW, + 4, W < < 0 G δ W + W 0 < < + W GW δ ( W ) 6.4 omuer lo 77

Semcouc hyscs evces: sc rcles, r eo her 6 (e le lk) 78

Semcouc hyscs evces: sc rcles, r eo her 7 her 7 7. G l where 0059. 5. 0 0 We () 0 5 () ( ) v () 0 5 0 6 0 7 0 8 0575. 065. 0695. 0754. 0 8 () ( ) v 0 5 0 6 0 7 0 8 0754. 084. 0874. 09. 7. S: 5. 0 0 Ge:.40 GAs: 8. 0 6 G l 0059. () 4 7 0, 0 S: 065., Ge: 05., GAs: 0. () 6 6 50, 50 S: 0778., Ge: 096., GAs: 5. 7 7 0, 0 S: 084., Ge: 04., GAs: 8. 7. omuer lo 7.4 omuer lo 7.5 () -se: G kl -se: 5 50 G 0059. l 0 5. 0 094. e G kl 7 0 G 0059. l 0 5. 0 04070. e () 0. 94 + 0. 4070 0764. G l ( 0059). l 076.. 0 50 7 5 5 0 0 8

Semcouc hyscs evces: sc rcles, r eo her 7 () G e + 4 (. ). (. ) 7 8850 076 6. 0 9 7 0 50 046. µ m / 0 + 50 5 7 5 4 7. 8850. 076. 9 6. 0 50 0 00. µ m We hve e Ε m 5 0 + 50 7 7 5 9 5 4 6. 0 50 0. 460 7. 8. 850 Ε m 9. 0 4 / 4 7.6 () -se 6 0 G 0059. l 0 5. 0 065. e -se 6 0 G 0059. l 0 5. 0 065. e () 0. 65 + 0. 65 0706. Q Q / / () G l ( 0059). l 0705. 6 6 00 0 5. 0 4 7. 8. 85 0 0. 705 9 6. 0 0 054. µ m y symmery 054. µ m e Ε m 6 0 0 + 0 6 6 6 9 6 4 6. 0 0 054. 0 7. 8. 850 Ε m 4.760 4 / e 4 7.7 () k 9 0..8 0 e 0059. 84. 0 5 (-reo) e k 9 04. 0 e 08. 0059. 9.970 5 (-reo) G l Q / 84

Semcouc hyscs evces: sc rcles, r eo her 7 ( 0059). l 0690. 5 5 9.970 8. 40 0 5. 0 7.8 () GAs: 0., 8. 0 6 0. W 0 +. 05. Also 05. G l 05. 0. ( 0. 059) l G 05. 0. e 0059. () e 0. 0 059 (. ) 4.40 6 05. 04. 0 6 G 4 e +. 8850. 0. 6. 0 9 4 066. µ m / 4.40 + 04. 0 6 6 / () 05. 0096. µ m (e) e Ε m e 7.9 () () 9 6 4 6. 0 04. 0 066. 0. 8. 850 Ε m 55. 0 4 / ( 0059). l 065. 4 6 5 0 0 0 5. 0 G e + 4 (. ). (. ) 7 8850 065 0864. µ m 6. 0 9 0 0 6 G e + 4 / 0 + 0 5 6 5 (. ). (. ) 7 8850 065 00864. µ m e Ε m 6. 0 9 0 0 5 / 0 + 0 6 6 5 9 5 4 6. 0 0 0. 8640 7. 8. 850 4 Ε m 4. 0 4 / Q Q / / 85

Semcouc hyscs evces: sc rcles, r eo her 7 7.0 G l e k We c wre 00 l l l l l + 00 k + k l l 00 0. 40 0. 050 00 5 6 50 0 Q 9 9.8 0 04. 0 00. + Q 0. 059 00 4. 4 558 + 00 00 00 l l 5. 44. l y rl err 490 7. () G l ( 0059). l 08556. 0 50 0 7 7 5. 0 () % che, ssume h he che s ue o, where he mj eeece o emerure s ve y e k l l l l l l l l k l l k l 9 9 U l.80 04. 0 + k W l 9 9 U l.8 0 04. 0 + W 7 7 l 5 0 0 7 7 / l 5 0 0 79.897 88. 567 + k 79.897 88. 567 + k We c wre 867. + 867. + k k 0990.. 4.57 867. + 0059. so h. 4.90 k ( 0059). 00 We he 0.4 k 86

Semcouc hyscs evces: sc rcles, r eo her 7 7. () 0 6, G kl 6 0 G 0059. l 0 5. 0 047. e 0 5, 5 0 G 0059. l 0 5. 0 0877. e 0. 47 0. 877 00596. 7. () () G l ( 0059). l 0456. 6 0 0 0 5. 0 4 7. 8850. 0456. 9 6. 0 0 0.40 7 0 + 0 6 6 4 7. 8850. 0456. 9 6. 0 0 0 6.40 0 + 0 6 Q Q / / () e Ε m 9 6 7 6. 0 0.40 4 ( 7. ) 8. 850 Ε m 75. 0 / 7.4 Assume Slco, so k / e 7. 8. 85 0 0. 059 6. 0 / 4 9 9 6. 0 6760 5 /. G () 80 4, 0447. µ m ().0 6, 0. 0760 µ m 80 7, 0. 004577 µ m () 0747. () 0886. 096. Also 7. 8850. 6. 0 () 096. µ m () 078. µ m 0. 07 µ m () 00. 9 4 G + 7 80 Q 7 80 / 87

Semcouc hyscs evces: sc rcles, r eo her 7 () 067. 0677. 7.5 omuer lo 7.6 () () W G l ( 0059). l 067. S 6 5 00 0 5. 0 / + + e 4 + 7. 8850. 6. 0 9 W 7.0 9 + + Q U 6 5 0 0 6 5 00 W / 0, W 069. 0 4 8, W.48 0 4 + Ε m W 0, Ε m 94. 0 4 / 8, Ε 7.00 4 / m / () Also 0856. G / 4... + e + 7 8850 5856 6. 0 50 05. µ m 7 9 0 50 + 0 7 7 7 + e + G / 4... 7 8850 5856 6. 0 9 7 0 50 50 + 0 0050. µ m Also W + W 00. µ m. Ε m + ( 5856) W 00. 0 4 () Ε m 89. 0 5 / A W 7 7 7 7. 8. 850 0 44. 00. 0 4 4 4 Q Q / / 7.7 () G l ( 0059). l 0 50 0 7 7 5. 0 7.8 () G l ( 0059). l 50 5. 0 0 88

Semcouc hyscs evces: sc rcles, r eo her 7 We c wre 075. 50 e 0 0059. 5. 0 0 5. 0 075. e 50 0. 059 () W 4.80 5.40 7 G / 4... 9 5 6. 0 4.80 + e 7 8850 075 80. µ m Q / / e + 9 4 5 6. 0 7. 8. 850 4.80 Q 075. 574. 0 9 / 7.9 () elec che S + + + >> 44. so 4.4% che. U W / / () k G l k l k l k + G k l l + So we c wre hs s k l so kl ( 0. 059) l 7.95 m 7.0 () W A W W A W We / + + A A e A / + + e + + A A A + + 8 5 0 0 0754. 0 5. 0 8 6 0 0 089. 0 5. 0 G 8 5 G 6 / 5754. 0 + 0 0 8 6 5 589. 0 + 0 0 0059. l A 0059. l So we W( A) W( ) W A W. / 89

Semcouc hyscs evces: sc rcles, r eo her 7 () Ε A Ε j j ( A) ( ) + A W( A) + W 5754.. 589. 06. Ε A Ε j W W A A + + / A + + A A / + + + + G G + G + A A A + 5 G 8 6 / 0 589. 0 0 + 6 8 5 0 5754. 0 0 ( A) ( ) j 09. 7. 5 7 40 40 () ( 0059). l 0 5. 0 so 0766. / + Ε m e + 9 6 0 5 0 4 7 8 850. + Q.. 5 7 4040 5 7 40 + 40 / 0 9 90 0 + + 7. 77 7 () 6 7 4 04 0 0059. l 0 5. 0 086. 9 6 0 5 0. 4 7 8 850 +... Q 6 7 4040 6 7 + Q 40 40 whch yels + 8007. 7.8 7 7 4 04 0 0059. l 0 5. 0 0886. 9 6 0 5 0. 4 7 8 850 +.. whch yels + 456. 0570. 7. () We hve () 0 j ( 0) j () 0 j ( 0) j Q 7 7 4040 7 7 + Q 40 40 + + +. / / + / 90

Semcouc hyscs evces: sc rcles, r eo her 7 0, we (. ) + 0 4. () 0W 0 + 05. l so.. 4. 0. 059 l We c he wre 6 8. 0 e 05. 05. 8. 0 4. 6 ( 0. 059). 0 6 5. 0 5 7. 6 6 0 50 ( 0059. ) l 6 8. 0 0. / j + / j + So 0. + () 0. + 8. 6 + + / 7.4 / 8 5 0 0 0 5. 0 e + + ( 0059). l 0754. >>, we hve + 9 4 5 /. 60 ( 7. ) 8. 850 0 880 7 /. +, 687. 0 9 / 0,.770 9 / A 60 4, he, 4. 0, 66. he reso requecy s ve y π so h, 67. z 0,.6 z 7.5 e Ε m + juco, so h / + e / e Ε m + Assum h <<, he 9

Semcouc hyscs evces: sc rcles, r eo her 7 Ε m 7. 8. 85 0 0 9 e 6. 0 ( 0) 4 6 4. 0 7 7.6 0W 0 + whch yels 9 We c wre G l.. 9 ( 0059. ) l 0 5. 0 We lso hve 5. 0 9 66. 0 / j 4 A 55. 0 so / 9 e 66. 0 + + Whch ecomes 4.050 7 6. 9 4 0 ( 7. ) 8. 850 9 + + 9 7 7.460 4.050 +., he y ero we 7.7 () 9.90 4 06. 89. 0 5 G l ( 0059). l 0 50 0 5 4 5. 0 () Also 0557. 00 G e + 4 (. ). (. ) 7 8850 0557 6. 0 4 0 50 9 5. 0 6 G e + / 0 + 50 5 4 5 4 (. ). (. ) 7 8850 0557 6. 0 9 50 0 5.660 4 0 µ m, we hve 4 5 50 0. / 0 + 50 4 4 5 4 7. 8850. + 6. 0 9 0 + 50 whch ecomes 6 7 90 70 + We 70. 4 4 5 Q Q / / Q / 7.8 A + juco wh 0 4, () A oe-se juco ssume >>, he so e / 9

Semcouc hyscs evces: sc rcles, r eo her 7 7 ( 4 8850 4. 500 ). 9 4 6. 0 0 whch yels 9 () G so 4 500 4 0 G 6 0 05. µ m Ε m + ( 9) W 50. 50 4 7.9 () Ε m 7.70 4 / ( 0059) 0796.. l A A 0 50 8 5 5. 0 0 / e + + 9 4 60 ( 7) 8 850 5 50... + 8 5 0 50 8 5 + Q 0 50 6 / 5 4.0 50 + 0, 4., 05. 6, 089. We c wre A + + e / he + juco + A e so h A e We hve 0, 7.690 6, 660., 6, 584. 0 4 We Ae G 5 9 4 50 60. ( 7. ) 8850. J 4 4 584. 0 G 6 so h 5 5 4.96 0 50, srh le y m+ 4 584. 0 m 6 A 0, 7.690 4 584. 0 7.690 6 +, 0, 4 584. 0 0 + 7.690 6 whch yels 0790. 9

Semcouc hyscs evces: sc rcles, r eo her 7 0796. () 8 6 0 6 0 0059. l 0 5. 0 0860. 9 4 60 ( 7) 8 850 5 50... + 5 4.6890 50 + 0, 69., 74. 6,. 5 8 6 0 60 / 0 + 60 8 6 7.0. 0. 0 / 5 A 0 () oe-se juco / 7 e + where s he o cocero he low-oe reo. We hve + 095. + 005. 00. 0 7. 9 4 6 0 ( 7).. 8. 85 0 () whch yels.040 7 () G l where s he o cocero he hh-oe reo. So Q / 0. 95 0. 059 l whch yels 7. omuer lo.040 7 5. 0 9.80 8 7. () G l -reo Ε ρ( ) e Ε e + 0 We hve e Ε0 < < 0 Ε e + -reo, 0 < < Ε ρ( ) e e Ε + -reo, < < Ε ρ( ) e e Ε + We hve Ε 0, he e so h < < Ε e We lso hve 94

Semcouc hyscs evces: sc rcles, r eo her 7 Ε Ε e e + e, 0 < <, e Ε e 7. φ() ρ() () () Ε < < µ m, ρ() + e So Ε e e Ε + A µ m, Ε 0 So 0 e e + e Ε + Ε A 0, Ε() 0 µ m, so e Ε 0 + 0 4 () whch yels 9 5 6. 0 50 4 0 4 ( 7. ) 8. 850 Ε() 0 7.70 4 / ue o oel erece s e φ Ε + z e + + e φ 0, he e e 0 + we c wre z e φ + A µ m 9 5 6. 0 50 4 φ + 0 4 7 (. ) 8850. φ 86. oel erece cross he rsc reo 4 4 φ Ε() 0 7.700 φ 55. y symmery, oel erece cross he - reo sce chre reo s lso 86.. he ol reverse-s vole s he ( 86. ) + 55.. 7.4 () he lerly re juco, ρ() e, Ε ρ( ) e Ε z e e + A +, Ε0 So e 0 + e G e Ε () e φ() z Ε + Se φ 0, he 0 e e + + e e φ() G + 95

Semcouc hyscs evces: sc rcles, r eo her 7 7.5 We hve h e + he 7. 0 9 whch yels / 9 4 6. 0 ( 7. ) 8. 85 0 ( 0. 7 5. + ). 0 0 4 96

Semcouc hyscs evces: sc rcles, r eo her 8 her 8 8. he wr s e e S k e e k e S e k S e k () () l G k e J e 0 59.9 m 60 m 00 9. m 0m 8. e e S k we c wre hs s e + e k S so h k l + e S reverse s, s eve, so 090., we hve S 0059. l 090. 59.6 m 8. omuer lo 8.4 he cross-secol re s 00 A 50 J 0 We hve 4 J J e S 065. 0 J S e 0059. so h J.50 0 A/ S We c wre J e + S Q \ τ τ We w τ 00. + τ τ 5 7 50 5 0 + 7 7 50 50 7.070 00. 7.070 + 4.470 whch yels 4.4 J.5 0 0 60 9 50 0.. S 5 0 + 7 7 ( 4.4) 50 50 We 7.0 4 0. 0 6 0